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2's Complement Representation for Signed Integers
Definition
Calculation of 2's Complement
Addition
Subtraction
Multiplication
Division
Sign Extension
Other Signed Representations
Notes
Definition
Property
Two's complement representation allows the use of binary arithmetic operations on signed
integers, yielding the correct 2's complement results.
Positive Numbers
Positive 2's complement numbers are represented as the simple binary.
Negative Numbers
2. Negative 2's complement numbers are represented as the binary number that when added to
positive number of the same magnitude equals zero.
Integer
2's Complement
Signed Unsigned
5 5 0000 0101
4 4 0000 0100
3 3 0000 0011
2 2 0000 0010
1 1 0000 0001
0 0 0000 0000
-1 255 1111 1111
-2 254 1111 1110
-3 253 1111 1101
-4 252 1111 1100
-5 251 1111 1011
Note: The most significant (leftmost) bit indicates the sign of the integer; therefore it is sometimes calle
the sign bit.
If the sign bit is zero,
then the number is greater than or equal to zero, or positive.
If the sign bit is one,
then the number is less than zero, or negative.
Calculation of 2's Complement
To calculate the 2's complement of an integer, invert the binary equivalent of the number by changin
all of the ones to zeroes and all of the zeroes to ones (also called 1's complement), and then add one
For example,
0001 0001(binary 17) 1110 1111(two's complement -17)
NOT(0001 0001) = 1110 1110 (Invert bits)
1110 1110 + 0000 0001 = 1110 1111 (Add 1)
3. 2's Complement Addition
Two's complement addition follows the same rules as binary addition.
For example,
5 + (-3) = 2 0000 0101 = +5
+ 1111 1101 = -3
0000 0010 = +2
2's Complement Subtraction
Two's complement subtraction is the binary addition of the minuend to the 2's complement of the
subtrahend (adding a negative number is the same as subtracting a positive one).
For example,
7 - 12 = (-5) 0000 0111 = +7
+ 1111 0100 = -12
1111 1011 = -5
2's Complement Multiplication
Two's complement multiplication follows the same rules as binary multiplication.
For example,
(-4) × 4 = (-16) 1111 1100 = -4
× 0000 0100 = +4
1111 0000 = -16
2's Complement Division
Two's complement division is repeated 2's complement subtraction. The 2's complement of the divisor
calculated, then added to the dividend. For the next subtraction cycle, the quotient replaces the
4. dividend. This repeats until the quotient is too small for subtraction or is zero, then it becomes the
remainder. The final answer is the total of subtraction cycles plus the remainder.
For example,
7 ÷ 3 = 2 remainder 1 0000 0111 = +7 0000 0100 = +4
+ 1111 1101 = -3 + 1111 1101 = -3
0000 0100 = +4 0000 0001 = +1 (remainder)
Sign Extension
To extend a signed integer from 8 bits to 16 bits or from 16 bits to 32 bits, append additional bits on the
side of the number. Fill each extra bit with the value of the smaller number's most significant bit (the sig
bit).
For example,
Signed Integer 8-bit Representation 16-bit Representation
-1 1111 1111 1111 1111 1111 1111
+1 0000 0001 0000 0000 0000 0001
Other Representations of Signed Integers
Sign-Magnitude Representation
Another method of representing negative numbers is sign-magnitude. Sign-magnitude
representation also uses the most significant bit of the number to indicate the sign. A negative
number is the 7-bit binary representation of the positive number with the most significant bit set
one. The drawbacks to using this method for arithmetic computation are that a different set of
rules are required and that zero can have two representations (+0, 0000 0000 and -0, 1000 0000)
Offset Binary Representation
A third method for representing signed numbers is offset binary. Begin calculating a offset binar
code by assigning half of the largest possible number as the zero value. A positive integer is the
absolute value added to the zero number and a negative integer is subtracted. Offset binary is
popular in A/D and D/A conversions, but it is still awkward for arithmetic computation.
For example,
Largest value for 8-bit integer = 28 = 256
6. Conversion to Two's Complement
Arithmetic with Two's Complement
Why Inversion and Adding One Works
Two's complement is not a complicated scheme and is not well served by anything lengthly. Therefore, after this introduction, which explains
what two's complement is and how to use it, there are mostly examples.
Two's complement is the way every computer I know of chooses to represent integers. To get the two's complement negative notation of an
integer, you write out the number in binary. You then invert the digits, and add one to the result.
Suppose we're working with 8 bit quantities (for simplicity's sake) and suppose we want to find how -28 would be expressed in two's
complement notation. First we write out 28 in binary form.
00011100
Then we invert the digits. 0 becomes 1, 1 becomes 0.
11100011
Then we add 1.
11100100
That is how one would write -28 in 8 bit binary.
Conversion from Two's Complement
Use the number 0xFFFFFFFF as an example. In binary, that is:
1111 1111 1111 1111 1111 1111 1111 1111
What can we say about this number? It's first (leftmost) bit is 1, which means that this represents a number that is negative. That's just the
way that things are in two's complement: a leading 1 means the number is negative, a leading 0 means the number is 0 or positive.
To see what this number is a negative of, we reverse the sign of this number. But how to do that? The class notes say (on 3.17) that to
reverse the sign you simply invert the bits (0 goes to 1, and 1 to 0) and add one to the resulting number.
The inversion of that binary number is, obviously:
0000 0000 0000 0000 0000 0000 0000 0000
Then we add one.
0000 0000 0000 0000 0000 0000 0000 0001
So the negative of 0xFFFFFFFF is 0x00000001, more commonly known as 1. So 0xFFFFFFFF is -1.
Conversion to Two's Complement
Note that this works both ways. If you have -30, and want to represent it in 2's complement, you take the binary representation of 30:
0000 0000 0000 0000 0000 0000 0001 1110
Invert the digits.
7. 1111 1111 1111 1111 1111 1111 1110 0001
And add one.
1111 1111 1111 1111 1111 1111 1110 0010
Converted back into hex, this is 0xFFFFFFE2. And indeed, suppose you have this code:
#include <stdio.h>
int main() {
int myInt;
myInt = 0xFFFFFFE2;
printf("%dn",myInt);
return 0;
}
That should yield an output of -30. Try it out if you like.
Arithmetic with Two's Complement
One of the nice properties of two's complement is that addition and subtraction is made very simple. With a system like two's complement,
the circuitry for addition and subtraction can be unified, whereas otherwise they would have to be treated as separate operations.
In the examples in this section, I do addition and subtraction in two's complement, but you'll notice that every time I do actual operations with
binary numbers I am always adding.
Example 1
Suppose we want to add two numbers 69 and 12 together. If we're to use decimal, we see the sum is 81. But let's use binary instead, since
that's what the computer uses.
1 1
Carry Row
0000 0000 0000 0000 0000 0000 0100 0101
(69)
+ 0000 0000 0000 0000 0000 0000 0000 1100
(12)
0000 0000 0000 0000 0000 0000 0101 0001
(81)
Example 2
Now suppose we want to subtract 12 from 69. Now, 69 - 12 = 69 + (-12). To get the negative of 12 we take its binary representation, invert,
and add one.
0000 0000 0000 0000 0000 0000 0000 1100
Invert the digits.
8. 1111 1111 1111 1111 1111 1111 1111 0011
And add one.
1111 1111 1111 1111 1111 1111 1111 0100
The last is the binary representation for -12. As before, we'll add the two numbers together.
1111 1111 1111 1111 1111 1111 1 1
Carry Row
0000 0000 0000 0000 0000 0000 0100 0101
(69)
+ 1111 1111 1111 1111 1111 1111 1111 0100
(-12)
0000 0000 0000 0000 0000 0000 0011 1001
(57)
We result in 57, which is 69-12.
Example 3
Lastly, we'll subtract 69 from 12. Similar to our operation in example 2, 12 - 69 = 12 + (- 69). The two's complement representation of 69 is
the following. I assume you've had enough illustrations of inverting and adding one.
1111 1111 1111 1111 1111 1111 1011 1011
So we add this number to 12.
111
Carry Row
0000 0000 0000 0000 0000 0000 0000 1100
(12)
+ 1111 1111 1111 1111 1111 1111 1011 1011
(-69)
1111 1111 1111 1111 1111 1111 1100 0111
(-57)
This results in 12 - 69 = -57, which is correct.
Why Inversion and Adding One Works
Invert and add one. Invert and add one. It works, and you may want to know why. If you don't care, skip this, as it is hardly essential. This is
only intended for those curious as to why that rather strange technique actually makes mathematical sense.
Inverting and adding one might sound like a stupid thing to do, but it's actually just a mathematical shortcut of a rather straightforward
computation.
Borrowing and Subtraction
9. Remember the old trick we learned in first grade of "borrowing one's" from future ten's places to perform a subtraction? You may not, so I'll
go over it. As an example, I'll do 93702 minus 58358.
93702
- 58358
-------
Now, then, what's the answer to this computation? We'll start at the least significant digit, and subtract term by term. We can't subtract 8 from
2, so we'll borrow a digit from the next most significant place (the tens place) to make it 12 minus 8. 12 minus 8 is 4, and we note a 1 digit
above the ten's column to signify that we must remember to subtract by one on the next iteration.
1
93702
- 58358
-------
4
This next iteration is 0 minus 5, and minus 1, or 0 minus 6. Again, we can't do 0 minus 6, so we borrow from the next most significant figure
once more to make that 10 minus 6, which is 4.
11
93702
- 58358
-------
44
This next iteration is 7 minus 3, and minus 1, or 7 minus 4. This is 3. We don't have to borrow this time.
11
93702
- 58358
-------
344
This next iteration is 3 minus 8. Again, we must borrow to make thi 13 minus 8, or 5.
1 11
93702
- 58358
-------
5344
This next iteration is 9 minus 5, and minus 1, or 9 minus 6. This is 3. We don't have to borrow this time.
1 11
93702
- 58358
-------
35344
So 93702 minus 58358 is 35344.
Borrowing and it's Relevance to the Negative of a Number
10. When you want to find the negative of a number, you take the number, and subtract it from zero. Now, suppose we're really stupid, like a
computer, and instead of simply writing a negative sign in front of a number A when we subtract A from 0, we actually go through the steps of
subtracting A from 0.
Take the following idiotic computation of 0 minus 3:
1 11 111 1111
000000 000000 000000 000000 000000
- 3 - 3 - 3 - 3 - 3
------ ------ ------ ------ ------
7 97 997 9997
Et cetera, et cetera. We'd wind up with a number composed of a 7 in the one's digit, a 9 in every digit more significant than the 100's place.
The Same in Binary
We can do more or less the same thing with binary. In this example I use 8 bit binary numbers, but the principle is the same for both 8 bit
binary numbers (chars) and 32 bit binary numbers (ints). I take the number 75 (in 8 bit binary that is 010010112) and subtract that from zero.
Sometimes I am in the position where I am subtracting 1 from zero, and also subtracting another borrowed 1 against it.
1 11 111 1111
0000000 0000000 0000000 0000000 0000000
0 0 0 0 0
- - - - -
0100101 0100101 0100101 0100101 0100101
1 1 1 1 1
------- ------- ------- ------- -------
--- --- --- --- ---
1 01 101 0101
1111111
11111 111111 1111111 1
0000000 0000000 0000000 0000000
0 0 0 0
- - - -
0100101 0100101 0100101 0100101
1 1 1 1
------- ------- ------- -------
--- --- --- ---
10101 110101 0110101 1011010
1
If we wanted we could go further, but there would be no point. Inside of a computer the result of this computation would be assigned to an
eight bit variable, so any bits beyond the eighth would be discarded.
11. With the fact that we'll simply disregard any extra digits in mind, what difference would it make to the end result to have subtracted 01001011
from 100000000 (a one bit followed by 8 zero bits) rather than 0? There is none. If we do that, we wind up with the same result:
11111111
100000000
- 01001011
----------
010110101
So to find the negative of an n-bit number in a computer, subtract the number from 0 or subtract it from 2n. In binary, this power of two will be
a one bit followed by n zero bits.
In the case of 8-bit numbers, it will answer just as well if we subtract our number from (1 + 11111111) rather than 100000000.
1
+ 11111111
- 01001011
----------
In binary, when we subtract a number A from a number of all 1 bits, what we're doing is inverting the bits of A. So the subtract operation is
the equivalent of inverting the bits of the number. Then, we add one.
So, to the computer, taking the negative of a number, that is, subtracting a number from 0, is the same as inverting the bits and adding one,
which is where the trick comes from.
Thomas Finley 2000
